A strictly distribution-free test has been proposed for testing that several independent pairs of random variables have locations as specified. Under the null hypothesis, the test statistic has asymptotically a chi-square distribution with degrees of freedom two. The test has been shown to be unbiased and consistent against reasonable classes of alternatives. Asymptotic power and efficiency have been examined.
Shoutir Kishore Chatterjee. "A Bivariate Sign Test for Location." Ann. Math. Statist. 37 (6) 1771 - 1782, December, 1966. https://doi.org/10.1214/aoms/1177699165